Dynamics of global atmospheric CO 2 concentration from 1850 to 2010 : a linear approximation

concentration through this time period may be properly approximated as a linear system. We demonstrate this hypothesis by deriving a linear box-model to describe carbon exchanges between the atmosphere and the surface reservoirs under the influence of disturbances such as anthropogenic CO2 emissions and global temperature changes. We show that the box model accurately simulates the observed atmospheric CO2 10


Introduction
Anthropogenic CO 2 emissions from fossil-fuel usage and land-use changes have been almost exponentially increasing since the Industrial Revolution (Fig. 1).Their accumulation in the atmosphere appears to be changing Earth's climate (IPCC, 2007) reached because only 41-45 % of the CO 2 emitted between 1850 and 2010 remained in the atmosphere while the rest was sequestered by lands and oceans (Jones and Cox, 2005;Canadell et al., 2007;Raupach et al., 2008;Knorr, 2009) (Fig. 1).This largely constant ratio, generally referred to as the "airborne fraction" (denoted as "γ" in this paper), was conventionally used to evaluate the efficiency of global carbon sinks (carbon sequestered by lands and oceans) in assimilating the extra CO 2 from the atmosphere (Jones and Cox, 2005;Canadell et al., 2007).A few recent studies found that the airborne fraction can also be influenced by trends and variations in anthropogenic emissions and other extrinsic forcing events (e.g., volcanic eruptions), and thus may not be an ideal indicator for monitoring changes in the carbon sink efficiency (Knorr, 2009;Gloor et al., 2010;Frölicher et al., 2013).Although these previous studies mainly focused on explaining the constancy of the airborne fraction from the perspective of global carbon-cycle dynamics, we can look at the problem from a different perspective.For instance, we can use the stable airborne fraction to examine the dynamic characteristics of the carbon cycle.This paper is dedicated to address the latter question.In particular, we propose that the responses of atmospheric CO 2 concentration to the disturbances of anthropogenic CO 2 emissions since 1850s can be properly approximated by a linear dynamic system with largely constant parameters.
Our hypothesis is based on two important observations.First, it is a basic feature of a stable linear system that its responses to an exponentially increasing forcing [e.g., exp(αt), α > 0, t ≥ 0] will approach the input signal with a constant ratio.Indeed, because exponential functions are invariant (to the extent of a constant factor) to differential operators, the effects of a linear system on an exponential input are simply to delay the signal or/and to regulate its magnitude (Naylor and Sell, 1982).Because the intrinsic responses of a stable linear system must be decaying with time, the output from the system will be increasingly dominated by an exponential signal proportional to the input [i.e., ∝ exp(αt)].
Second, a nonlinear dynamic system can be linearly approximated around a steady point within a neighborhood in its state space (Khalil, 2001) carbon cycle, the existence of such a (quasi) steady state is evident in that the atmospheric CO 2 concentration (and the corresponding global climatology) had been stable for thousands of years before the industrial era (IPCC 2007).Therefore, the observed simple proportional relationship between the atmospheric CO 2 concentrations and the increasing CO 2 emissions suggest that recent changes of the global carbon cycle may be still within the linear neighborhood of the system's last steady state.
Another disturbance that this study intends to examine is the impact of climate changes on the atmospheric CO 2 dynamics.Global surface temperature has increased by ∼ 1 • C since the beginning of the 20th century (Hansen et al., 1999;Brohan et al., 2006).Given the tight coupling between temperature and the carbon cycle (Keeling et al., 1995;Joos et al., 1999Joos et al., , 2001;;Lenton, 2000;Rafelski et al., 2009), the warming alone may release a large amount of CO 2 from the land and the oceans into the atmosphere, redistributing carbon among these reservoirs.The impacts of temperature changes on the carbon system have different dynamic characteristics than those of CO 2 emissions and need to be analyzed separately (see below).
We demonstrate our analytical framework by developing a simple two-box model that represents carbon exchanges between the atmosphere and the surface (i.e.land and ocean) reservoirs.Although such a "toy" model only represents a first-order approximation of the global carbon cycle, important characteristics of the atmospheric CO 2 dynamics can still be learned from it.Throughout the analysis we compare the results obtained from the two-box model to those from the more advanced Bern model (Siegenthaler and Joos, 1992;Enting et al., 1994;IPCC 1996IPCC , 2001)), which is slightly revised and recalibrated in this study (see the Appendix).The Bern model couples the atmosphere with a process-based ocean biogeochemical scheme (Siegenthaler and Joos, 1992;Shaffer and Sarmiento, 1995;Joos et al., 1999) and a multi-component terrestrial biosphere module (Siegenthaler and Oeschger, 1987).The global carbon-cycle processes described in the Bern model help us diagnose the biogeophysical mechanisms underlying the characteristics of the atmospheric CO 2 dynamics identified with our simple linear model.

Datasets
Annual atmospheric CO 2 concentration data from 1850 to 1960 are based on the ice core CO 2 records from Law Dome, Antarctica (Etheridge et al., 1996) and those between 1960 and 2010 are compiled from NOAA Earth System Research Laboratory (ESRL) (Keeling et al., 1995;Conway et al., 1994).We merged the data following the approach described in Le Quéré et al. (2009) and calculated annual CO 2 growth rate as the first-order difference of the yearly CO 2 concentrations.Long-term records of anthropogenic CO 2 emissions from fossil fuel burning and cement production are compiled by Boden et al. (2011) and those of land-use changes are from Houghton (2003), both downloaded from the Carbon Dioxide Information Analysis Center at Oak Ridge National Laboratory, TN, USA (http://cdiac.ornl.gov).Two sets of monthly surface temperature data are used, including GISTEMP from NASA Goddard Institute for Space Studies (Hansen et al., 1999) and the CRU-NCEP climate dataset (Sitch et al., 2008;Le Quéré et al., 2009), available from 1901 to the present with spatial resolutions of 0.5 × 0.5 (CRU-NCEP) or 1 × 1 (GISTEMP) degrees.Monthly time series of temperature are aggregated globally and over the tropics ( 24• N-24 • S), and smoothed with a 12 month running window to convert the monthly data to annual values.We calculated temperature anomalies relative to their 1901 to 1920 annual mean and assumed the 20-years mean temperature to be representative of temperature climatologies between 1850 and 1900.This assumption is reasonable as suggested by analysis of other long-term coarse-resolution temperature datasets (Jones et al., 2003;Brohan et al., 2006).

Derivation of the two-box model
This study considers only the "fast" carbon flows between the atmosphere and the (combined land and oceans) surface at time scales within hundreds of years (IPCC are inflows to the surface (e.g., through photosynthesis in green vegetation and the dissolution of CO 2 in the surface water) while carbon outflows from the surface (e.g., through respiration and the outgassing of the dissolved CO 2 ) are the inflows to the atmosphere.Human emissions of CO 2 , on the other hand, represent an "external" source of CO 2 to the system by rapidly releasing carbon (e.g., fossil fuel burning) from reservoirs that were formed over millions of years and by permanently altering the structure of land surface carbon pools (e.g., land-cover/land-use changes).Following our linearization proposition, we describe the dynamics of such a two-box carbon system using the following equations: where A and S denote carbon storages in the atmosphere and the surface reservoirs, respectively, and E is the accumulated anthropogenic CO 2 emissions since the industrial era.The three variables can be measured by the same unit of parts per million by volume (1 ppm =∼ 2.13 × 10 9 metric-ton carbon or GtC).The prime symbol (e.g., "E ") indicates that changes in a variable relative to its preindustrial steady-state level.The preindustrial emissions are assumed negligible so that E = E .The dot accent (e.g., " Ė ") indicates the first-order derivative with regard to time, such that Ė represents the annual rate of CO 2 emissions (ppm yr −1 ).The positive constant parameters α A and α S (yr −1 ) describe the decaying rates of corresponding carbon anomalies.Their reciprocals (i.e., Because mass (carbon) is conserved in the two-box model, Eq. ( 1a) and (1b) are not independent.Adding the two equations together leads to Equation ( 1c) has a clear physical meaning that the anthropogenically emitted CO 2 either resides in the atmosphere or in the surface reservoirs (i.e., the land and the oceans).Substituting this relationship into Eq.( 1a) to replace S , we obtain Therefore, the dynamics of atmospheric CO 2 represented by the two-box model is determined by an ordinary differential equation of A under the disturbances of anthropogenic emissions (E and Ė ) and the changing climate (T ).

Model determination and evaluation
We want to determine the parameters of Eq. ( 2a) with observational records of A , T , and E , and evaluate how well the model captures the observed atmospheric CO 2 dynamics.In order to construct a regression model from Eq. ( 2a), we rearrange the equation as follows: where Ė − Ȧ represents the strength of annual carbon sinks.However, because A ≈ γ •E (Fig. 1), where γ is the airborne fraction, the "collinearity" between the two regressors prevents us from determining the coefficients associated with them separately  (Chatterjee and Hadi, 2006).Indeed, substituting the airborne-fraction relationship into Eq.(2b) leads to which implies that only β T and a combination of α A and α S can be estimated from the observations.We may reach the above argument from another perspective.Figure 1 shows that the anthropogenic CO 2 emissions (E and Ė ) can be approximated by exponential functions.Let E = exp(α E t), and it follows that Ė = α E • exp(α E t).By applying the airborne fraction relationship (A ≈ γ • E and Ȧ ≈ γ • Ė ) and neglecting the influence of temperature (β T T ) for now, we obtain The term on the left-hand side of Eq. ( 2d) is the same as the (regression) coefficient associated with the A in Eq. (2c).Equation (2d) suggests that this coefficient is closely related to the exponential disturbances (α E ) to the system.Because α E > 0 and γ is about 0.41-0.45(Fig. 1), it is follows that (1/γ − 1) > 1 and α A > α S (or τ A < τ S ).
Given that the response time of atmospheric CO 2 is at the order of 10 years (Revelle and Suess, 1957), we choose τ A to be 12 years (α A ≈ 0.083 yr −1 ) and subsequently estimate τ S to be ∼ 34 years (α S ≈ 0.029 yr −1 ).
The estimation of the β T parameter in Eq. (2a) requires the choice of a large-scale temperature index that is representative of climate change and closely related with global carbon cycle.Previous studies showed that the land surface air temperature in the tropics (24S-24N) are most strongly coupled with interannual variations in the growth rate of atmospheric CO 2 by a sensitivity (β T ) of ∼ 1.64 ppm yr −1 • C −1 (Adams and Piovesan, 2005;Wang et al., 2013).Here we found that the same temperature-CO 2 coupling may also operate at longer time scales.Indeed, because the system is linear, variations in A , T , E , and their derivatives over different time scales must satisfy Eq. (2a) separately.Because the interannual variations ("IAV") in the emissions (both Ė and E ) and the atmospheric CO 2 concentration (A ) are relatively small (Fig. 1), neglecting them in Eq. (2a) leads to which is the same linear relationship as previous reported (Wang et al., 2013).
With the model parameters determined, we use the two-box model to simulate the changes of atmospheric CO 2 concentration between 1850 and 2010 from historical records of temperature and CO 2 emissions (Fig. 2).The simulated results follow the evolution of the observed atmospheric CO 2 concentration to a high degree of accuracy, capturing more than 96 % of the variability (i.e., r 2 > 0.96) of the latter (Fig. 2).
The standard deviations (σ) of the differences between simulated values and those measured accurately since 1960 are∼ 0.9 ppm for the atmospheric CO 2 concentration and∼ 0.4 ppm for its growth rate, respectively (Fig. 2).These results are highly comparable to those simulated with the Bern model (Fig. 2) or the more sophisticated climate-carbon models reported in the literature (e.g., Joos et al., 1999;Lenton, 2000;Friedlinstein et al., 2006), strongly supporting our hypothesis that the atmospheric CO 2 dynamics in the past one and half centuries can be properly approximated with linear models.

Disturbance-response functions
A question of importance to the carbon-cycle and the climate science is how fast (and how much of) the anthropogenically emitted CO 2 will be absorbed by the land and the oceans.We address this question by analyzing the responses of the box model to some idealized disturbances (Fig. 3).For the simplicity of the analysis, we assume that changes in temperature are independent from the disturbances of anthropogenic CO 2 emissions.
We first check the model's responses to an impulse disturbance of anthropogenic CO 2 emissions.Shown in Fig. 3, the initial atmospheric CO 2 anomaly decays relatively fast, as 60-70 % of the emitted CO 2 is absorbed by the surface reservoirs within 20 years of the disturbance.However, the rate of carbon assimilation by the land and the oceans significantly slows down in the following decades and eventually becomes neutral as the system approaches steady-state.In the end, 15-25 % of the simulated CO 2 anomaly will likely stay in the atmosphere for thousands of years (Fig. 3).These results are generally consistent with the findings from fully coupled climate-carbon models (Cao et al., 2009;Archer et al., 2009;Joos et al., 2013).
The impulse-response functions of the linear box models can be analytically characterized.For the two-box model of Eq. (2a), when the system approaches a (new) steady state after the disturbance, all the time derivatives ( Ė and Ȧ ) will be zero.Assuming that temperature does not change during the process, we easily obtain the steady state of A as or more generally where the mass-conservation relationship represented by Eq. ( 1c) is used in the derivation.Therefore, the extra CO 2 added to the "fast" carbon cycle by anthropogenic emissions will be partitioned between the atmosphere and the surface corresponding to the response times (τ) of the reservoirs, respectively.Because τ S > τ A (see Eq. the related discussions), a majority of the emitted CO 2 will eventually be absorbed by the surface carbon reservoirs (Fig. 3).In other words, the long-term fate of the CO 2 emitted into the atmosphere is largely determined by the response times of the surface reservoirs (Revelle and Suess, 1957).
The rates at which the atmospheric CO 2 anomaly decays are determined by the solutions (i.e., eigenvalues) to the characteristic equation of the system.For a two-box system like Eq. ( 2a), the problem is particularly simple because the only eigenvalue (λ) is and the solution of Eq. ( 2a) is therefore A helpful observation of Eq. ( 5a) is that, when t 1/(α A + α S ), the solution can be approximated by That is, A initially decays at a maximum rate of α A as if the capacity of the surface carbon reservoir were unlimited (i.e., α S = 0).Next we consider the system's responses to disturbances induced by changes in surface temperatures.Unlike anthropogenic CO 2 emissions, changes in temperature do not add additional CO 2 to the "fast" carbon cycle but only re-distribute carbon between the atmosphere and the surface (Eq.1a and b), and so the system will recover to its initial steady state once the temperature anomaly is removed.However, increases in temperature are persistent under climate-change scenarios.Therefore, we examine the responses of atmospheric CO 2 to a step change in temperature, which is determined from Eq. (2a) as: Because α A > α S , for rough estimates we can also use Based on the estimated model parameters, therefore, atmospheric CO 2 may rise by ∼ 15 ppm for an increase of 1 • C in temperature within a few decades (Fig. 3).The relationships represented by Eqs.(3b), (5b), and (6b) can be easily generalized to higher-order systems, providing a convenient way to characterize the models' disturbance-response functions without fully solving the system equations.However, the uncertainties associated with these results -especially the long-term responses of atmospheric CO 2 -need to be emphasized.One key source of the uncertainties is that model's parameters are not fully determined by the observations of the global climate-carbon system.As discussed in Sect.4, the estimation of the model parameter α S depends on the choice of α A , which is only loosely constrained by the prior knowledge.It is possible for us to choose another pair of α A and α S so that the box model's disturbance response functions better approximate those of the Bern model.However, tuning the model in this fashion has only cosmetic effects to the results and does not reduce the associated uncertainties.In addition, in reality the climate system and the global carbon cycle are not independent but tightly coupled.Therefore, a comprehensive assessment of the long-term fate of anthropogenic CO 2 emissions in the atmosphere must account for the effects of the associated changes in global temperature, which is beyond the scope of this study and demands dedicated investigations in the future.

Biogeophysical implications
The biogeophysical implications of the above modeling results need additional discussions.To illustrate, we further rearrange Eq. ( 2c) to obtain On the left-hand side of the equation, the term " Ė − Ȧ " is usually used to measure the net strength of annual global carbon sinks.However, because the warming temperature also releases carbon on from the surface into the atmosphere (β T • T ), this extra source of CO 2 has to be absorbed by the global carbon sinks.By accounting for the effects of temperature changes, the term " Ė − Ȧ + β T • T " thus define a gross global carbon sinks.Examining Eq. ( 7) with the observational data shows that both the net and the gross carbon sinks have been steadily increasing in response to the rising atmospheric CO 2 concentration in the past 160 years, reaching ∼ 2.5 and ∼ 4.0 ppm yr −1 respectively in 2010 (Fig. 4).
Because the efficiency of sea-air carbon exchanges is constrained by the buffering effects of the surface biogeochemistry (Revelle and Suess, 1957), ocean carbon uptakes simulated by the Bern model account for ∼ 50 % of the net global sinks, while the other 50 % is contributed by land processes, consistent with the literature (Canadell et al., 2007;Le Quéré et al., 2009).Therefore, Fig. 4 suggests that the increasing atmospheric CO 2 concentration must have promoted carbon assimilation by the terrestrial biosphere (Ballantyne et al., 2012), most likely through the CO 2 fertilization effect (Körner and Arnone, 1992;Oechel et al., 1994;Long et al., 1991Long et al., , 2004) ) and the associated ecological changes (Keenan et al., 2013;Graven et al., 2013).Indeed, the gross CO 2 fertilization effects of terrestrial vegetation may be higher than previously thought: because the surface warming rapidly releases a proportion of the assimilated carbon back to the atmosphere (Fig. 4) (Piao et al., 2008;Wang et al., 2013), the increased turnover rate may have obscured the evaluation of the magnitude of the CO 2 fertilization effects, which we found in calibrating the Bern model (see the Appendix Shown in Fig. 4, the gross carbon sinks have a nearly direct linear relationship (with a constant slope ∼ 0.04 yr −1 ; r = 0.98) with the atmospheric CO 2 concentrations throughout the entire data period.In comparison, the relationship between the apparent carbon sinks and the CO 2 concentrations is slightly nonlinear, with its slope decreasing from ∼ 0.03 yr −1 in 1960 to ∼ 0.02 yr −1 in 2010.Note that the slope of these linear relationships can be largely interpreted as the efficiency of surface carbon reservoirs in sequestering annual CO 2 emissions.As such, although the gross carbon-sequestration rates of the surface reservoirs changed little, the net efficiency of the system may have been slowed by ∼ 30 % in the past five decades (Raupach et al., 2013).These results highlight the significant impacts of climate changes on the global carbon cycle.

Conclusions
Inspired by the fact that airborne fraction of anthropogenic CO 2 emissions has been remarkably stable in the observational records, we propose and demonstrate in this paper that the observed dynamics of the global atmospheric CO 2 concentration from 1850 to 2010 can be properly approximated as a linear system.In particular, we derived a simple box model to describe carbon exchanges between the atmosphere and the surface carbon reservoirs under the disturbances of anthropogenic CO 2 emissions as well as global temperature changes.We show that, with a few appropriately retrieved parameters, the model can successfully simulate the observed changes and variations of the atmospheric CO 2 concentration and its first-order derivative (i.e., CO 2 growth rate) across interannual to multi-decadal time scales.The results are highly comparable to those obtained with more sophisticated models in the literature, confirming that the simple linear model is capable in capturing the main features of atmospheric CO 2 dynamics in the past one and half centuries.
A distinct advantage of our linear modeling framework is that it allows us to analytically, and thus most directly, examine the dynamic characteristics of the (modeled) carbon-cycle system.Our analyses indicate that many of such characteristics are closely associated with the response times of the atmosphere and surface carbon reservoirs.For instance, the initial decaying rate of an impulse of CO 2 emitted into the atmosphere is mainly influenced by the response time of the atmosphere, but the proportion of the extra CO 2 that stays in the atmosphere at long-term time scales is determined by the ratio between the response times of the atmosphere and the surface reservoirs.Unfortunately, the collinearity exhibited by the observed time series of CO 2 emissions and atmospheric CO 2 concentrations has obscured the determination of the response times for individual surface reservoirs, inducing uncertainties of the estimated long-term responses of the global carbon system.In other words, although the steady CO 2 airborne fraction allows us to represent the carbon-system dynamics with a simple model, it prevents us to resolve further details of the surface carbon reservoirs.
Our model results also have important biogeophysical implications.They highlight that the responses of the global carbon cycle to recent anthropogenic and climatic disturbances are still within the resilience zone of the system, such that annual (gross) terrestrial and ocean carbon sinks linearly increases with the atmospheric CO 2 levels.On one hand, the elevated atmospheric CO 2 concentration must have enhanced land carbon uptakes through the "fertilization" effects and the associated ecological changes.On the other hand, however, the enhanced gross carbon uptakes are partially offset by the increases in global surface temperatures, which accelerate the release of carbon from the surface reservoirs into the atmosphere.As a result, the "net" efficiency of global land and oceans in sequestering atmospheric CO 2 may have slowed by ∼ 30 % since 1960s, although the airborne fraction of CO 2 emissions remains largely constant.
Finally and importantly, we emphasize that the linear approximation of the global carbon cycle discussed in this paper is conditioned on the pre-industrial (quasi) steady state of the system.The global climate-carbon system is clearly nonlinear beyond this scope (Archer et al., 2009), which can establish different steady states over glacial/interglacial time scales (Sigman and Boyle, 2000).A major concern stemming from climate change is that, because the post-industrial anthropogenic disturbances on the global carbon cycle are so strong and rapid, they may abruptly alter the pace at which the natural climate-carbon system evolves and drive the system into a different state at a drastically accelerated rate (IPCC 2001).Our results clearly indicate that the rising atmospheric CO 2 concentrations and the associated increases in global temperature have significantly intensified the global carbon cycle in the past one and half centuries.Although such intensification of the carbon system seems to be within the linear zone as of now, its resilience may be weakened, or lost, in the future.As the anthropogenic CO 2 emissions continue to increase and the global temperature continues to warm, scientists generally expect surface -in particular, terrestrial -carbon reservoirs to saturate and their CO 2 sequestration efficiency to decrease, such that the responses of the global carbon cycle to the anthropogenic disturbances will eventually deviate from their original path.With this concern regarded, the simple linear model developed in this study may serve as a convenient tool to monitor the early signs when the natural carbon system is pushed away (by anthropogenic emissions) from its linear zone.

Appendix A: Calibrations of the Bern carbon-cycle model
The Bern model is a coupled global carbon-cycle box model (Siegenthaler and Joos, 1992;Enting et al., 1994) that was used in previous IPCC Assessment Reports to study changes in atmospheric CO 2 concentration under different emission scenarios (IPCC 1996(IPCC , 2001)).It couples the High-Latitude Exchange/Interior Diffusion-Advection (HILDA) ocean biogeochemical model (Siegenthaler and Joos, 1992;Shaffer and Sarmiento, 1995;Joos et al., 1999) with an atmosphere layer and a multi-component terrestrial biosphere model (Siegenthaler and Oeschger, 1987).The HILDA model describes ocean biogeochemical cycling through two well-mixed surface layers in low and high latitudes, a well-mixed deep ocean in the high latitude and a dissipative interior ocean in the low latitude.Ocean tracer transport is represented by four processes: (1) eddy diffusion within the interior ocean (k, 3.2 × 10 port between the two surface layers as well as the down-welling in the polar deep ocean, (3) lateral exchange between the interior ocean and the well-mixed polar deep ocean (q, 7.5 × 10 −11 s −1 ); and (4) vertical exchange between the high-latitude surface layer and the deep polar ocean (u, 1.9 × 10 −6 m s −1 ) (Shaffer and Sarmiento, 1995).The effective exchange velocity between surface ocean layers and the atmosphere in both low and high latitudes is assumed to be the same (2.32 × 10 −5 m s −1 ) (Shaffer and Sarmiento, 1995).Ocean carbonate chemistry (e.g., the Revelle buffer factor) is based on the formulation given by Sarmiento et al. (1992).In addition, we implemented the influence of sea surface temperature on the partial pressure of dissolved CO 2 in seawater with a sensitivity of∼ 4.3 % Gordon and Jones, 1973;Takahashi et al., 1993;Joos et al., 2001).The changes in global mean sea-surface temperature (SST) is approximately 0.8-1.0 • C from 1850s to 2000s (Rayner et al., 2003;Brohan et al., 2006) slightly lower than that of the tropical land-based air temperature (∼ 1.0 • C) but with a trend resembling the latter (Rayner et al., 2003;Jones et al., 2003;Hansen et al., 2006).For simplicity, therefore, we used the long-term trend of the tropical land air as a proxy for the corresponding trend in global SST.The terrestrial biosphere in the Bern model is represented by four carbon compartments (ground vegetation, wood, detritus, and soil) with prescribed turnover rates and allocation ratios.The global net primary production (NPP), the influx to the biosphere, is assumed to be 60 GtC yr −1 at the pre-industrial level; and the effect of CO 2 fertilization on NPP (i.e., the β effect) is described with a logarithmic function with a β parameter of 0.38 (Enting et al., 1994).The original Bern model does not consider the effects of changing global temperatures on terrestrial ecosystem respiration, which have been suggested to play an important role in regulating the variability of the global carbon cycle at interannual to multi-decadal time scales (Wang et al., 2013;Rafelski et al., 2009).
Therefore, we implemented temperature's effects on terrestrial ecosystem respiration in the Bern model with an overall sensitivity (Q 10 ) of ∼ 1.5 (Lenton, 2000;Davidson and Janssens, 2006;Wang et al., 2013).We also changed the pre-industrial CO 2 concen- We calibrated the Bern model so that the model outputs fit the observed atmospheric CO 2 data most favorably.Because no major revisions were made to the ocean carbon cycle module (HILDA), we focused mainly on calibrating the biosphere module.
With the original biosphere model parameters, the simulated atmospheric CO 2 concentrations were found to be distinctly higher than observations, reaching ∼ 411 ppm in 2010.These results are induced because rising temperatures enhance respiration in the model, reducing the net land carbon sinks to an unrealistic ∼ 0.5 ppm yr −1 in 2010.To balance the temperature-enhanced respiration, we need to increase the β parameter from 0.38 to 0.64 to incorporate a higher rate of gross biosphere carbon uptake as enhanced by CO 2 fertilization (Long et al., 2004) and the associated ecological changes (Keenan et al., 2013).With the β parameter set at 0.64, the simulated global terrestrial NPP increased by 14 % from its pre-industrial level and reached ∼ 69 GtC yr −1 in 2010, which qualitatively agrees with recent estimates inferred from the isotope measurements (Welp et al., 2011).As such, the re-calibrated Bern model is able to simulate accurately the observed changes/variations in atmospheric CO 2 concentration and growth rate in the past 150 years (Fig. 2 of the main text).The simulated ocean and land components of global carbon sinks are also consistent with estimates found in previous studies (e.g., Canadell et al., 2007;Le Quéré et al., 2009).
Global Biogeochem. Cy., 7, 843-878, 1993. Wang, W., Ciais, P., Nemani, R., Canadell, J. G., Piao, S., and et al.: Variations in atmospheric CO 2 growth rates coupled with tropical temperature, P. Natl.Acad.Sci.USA, 110, 13061-13066, doi:10.1073/pnas.1219683110, 2013. Welp, L. R., Keeling, R. F., Meijer, H. A. J., Bollenbacher, A. F., Piper, S. C., andet al.: Inter-10 annual variability in the oxygen isotopes of atmospheric CO 2 driven by El Nino, Nature, 477, 579-582, 2011.In both annual and accumulative cases, CO . The full potential of anthropogenic CO 2 emissions for changing the climate has not yet Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | . In the case of the global 13959 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 2001).When human disturbances are absent, carbon outflows from the atmosphere 13961 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | are often referred to as the response time (or e-folding time) of the carbon reservoirs (IPCC 2001).T ( • C) denotes indices of global (or large-scale) surface temperatures and the coefficient β T (ppm yr −1 • C −1 ) represents the sensitivity of atmospheric CO 2 growth rate to temperature changes.The term β T T thus indicates the impacts of temperature increases on the global carbon cycle, which release CO 2 from the surface reservoirs to the atmosphere.Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

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Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | tration to 285 ppm in the Bern model to reflect the findings obtained from the observations (Fig. 4 of the main text).